What is expand (sqrt(3)-isqrt(3))^6 ?

The solution to expand (sqrt(3)-isqrt(3))^6 is 216i

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expand (sqrt(3)-isqrt(3))^6

expand

(3 - i 3)^{6}

216 i

Solution steps

(3 - i 3)^{6}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 3, b = - i 3

= i = 0 \sum 6 (i 6) (3)^{(6 - i)} (- i 3)^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} (3)^{6} (- i 3)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (3)^{5} (- i 3)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (3)^{4} (- i 3)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (3)^{3} (- i 3)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (3)^{2} (- i 3)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (3)^{1} (- i 3)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (3)^{0} (- i 3)^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (3)^{6} (- i 3)^{0} : 27

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (3)^{5} (- i 3)^{1} : - 162 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (3)^{4} (- i 3)^{2} : 405 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (3)^{3} (- i 3)^{3} : - 540 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (3)^{2} (- i 3)^{4} : 405 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (3)^{1} (- i 3)^{5} : - 162 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (3)^{0} (- i 3)^{6} : 27 i^{6}

= 27 - 162 i + 405 i^{2} - 540 i^{3} + 405 i^{4} - 162 i^{5} + 27 i^{6}

Simplify

27 - 162 i + 405 i^{2} - 540 i^{3} + 405 i^{4} - 162 i^{5} + 27 i^{6} : 216 i

= 216 i

cos (θ) (2 sin (θ) + 1)

\frac{5}{( s ^{2} + 25 ) ( 1 + 10 e ^{10 t} )}

- 2 \cdot (\frac{y}{3} + \frac{x}{6})

(1 - t) (0.3) + t (4.6)

\frac{2 d x}{( x ^{2} - 1 ) ^{2}}

What is expand (sqrt(3)-isqrt(3))^6 ?
The solution to expand (sqrt(3)-isqrt(3))^6 is 216i